I have found references (see for example https://dlmf.nist.gov/10.43) that the following equality holds $$\int e^{-x}K_{0}(x)dx=xe^{-x}(K_{0}(x)-K_{1}(x))$$ (here $K_{n}$ denotes the modified Bessel function of the second kind of order $n$).
Does this give me any help to find $$\int e^{-ax}K_{0}(x)dx,$$ with $a>0$?